Vorticity Analysis: Rotational vs. Irrotational Vortex

Lecture 11 11.1 Administration • Collect problem set. 11.2 2nd law of thermodynamics (Jim – You have a note for yourself here that reads “Fill in full derivation, this is a sloppy treatment”). For completeness, I feel obligated to go through the derivation of the 2nd law of thermodynamics as well. Kundu and Cohen actually do a god job with this one, so all I’m going to do is mimic their derivation. The change in entropy is given by 1 (11.1) T dS = de + p dα, α = ρ De Dα DS ⇒ T = +p (11.2) Dt Dt Dt Remember, the laws of thermodynamics are empirical. This definition of entropy (equation 11.1) “just is.” Also recall that de is the internal component, while p dα is the work component, and the differential form (equation 11.2) is achieved by dividing by dt and taking the limit. Next, recall that D ρ1 d x1 1 Dρ 1 dx = − (11.3) ⇒ =− 2 x2 dt dt Dt ρ Dt Hence T DS Dt = De p Dρ − 2 Dt ρ Dt (11.4) We already know that De Dt 1 p φ − �·q− �·u+ ρ ρ ρ = and Dρ = −ρ� · u Dt Thus, substitution yields DS Dt DS T Dt T = = 1 − �·q− ρ 1 − �·q+ ρ 1 (energy equation) (11.5) (continuity equation) (11.6) p φ p �·u+ + �·u ρ ρ ρ φ ρ (11.7) (11.8) We can expand this a bit more by noting that � � �q� 1 1 �· = q·� + �·q T T T d(1/x) dy = − x12 dx dy , this becomes �q� q 1 �· = − 2 · �T + � · q T T T �q� q 1 ⇒ − � · q = −� · − 2 · �T T T T Substituting this results into 11.8 yields �q� DS q φ ρ = −� · − 2 · �T + Dt T T T Now recall Fourier’s Law And again using (11.9) q = −K�T (11.10) (11.11) (11.12) (11.13) Substitution then yields �q� K φ(µ) DS = −� · + 2 �2 T + (11.14) Dt T T T Here the first term on the RHS is the entropy gain due to reversible heat transfer. The second term on the RHS is the entropy gain due to nonreversible heat conduction. The final term is the entropy gain due to viscous generation of heat. Because the 2nd law tells us that entropy production from irreversible processes must be positive, then we require K > 0 (thermal diffusivity), and µ > 0 (viscosity). ρ 11.3 Summary of equations (non-Boussinesq) Continuity Dρ + ρ� · u Dt = 0 (11.15) Momentum ρ Du Dt = ρg − �p + µ�2 u + µ �(� · u) 3 (11.16) Heat (assuming a linear equation of state) ρCv DT Dt = −p(� · u) + φ − � · q (11.17) State p = ρRT, or ρ = ρo (1 − αT (T − To ) + αS (S − So )) (11.18) The first equation is the ideal gas law, as commonly applied to the atmosphere. The second equation is the EOS for the ocean, where αT and αS are empirical. This also requires a salt 2 equation, typically something like DS Dt = γ� S. Entropy (Jim – K is not used here, you crossed out the ones where it was) µ > 0, K > 0 Hence, altogether, there are 6 (7 for ocean) equations and 6 (7) unknowns. 2 (11.19) 11.4 Summary of equations (Boussinesq) ρ = ρo + ρ� (x, y, z, t) (11.20) Continuity (to first order) �·u = 0 (11.21) = ρ� g − �p� + µ�2 u (11.22) Momentum (traditional to leave primes off) ρo Du Dt Heat (crap for atmosphere, not so great for ocean) DT Dt K�2 T, = K= k ρCp (11.23) State p = ρRT, ρ = ρo (1 − αT (T − To ) + αS (S − So )) or (11.24) Again, the first equation is the ideal gas law, and the second equation is a simple EOS for 2 the ocean. A salt equation, typically something like DS Dt = γ� S, is required. Entropy µ > 0, K > 0 11.5 (11.25) Summary of equations (Euler) (Jim – You have a note to yourself to clean this section up) µ = 0, � · u = 0 (usually, but not always) (11.26) Continuity (typically, can be full continuity equation) �·u = 0 (11.27) ρ� g − �p� (11.28) Momentum (traditional to leave primes off) ρo Du Dt = State (typically, can be any) ρ = ρo 3 (11.29) Figure 11.1: (fig:Lec11Circulation1) The circulation is the sum of the component of velocity tangent to C summing all around C. Figure 11.2: (fig:Lec11Circulation2) For steady, uniform flow, Γ = 0. 11.6 Vorticity – At last! Vortex motion is motion in circular streamlines. But, just because you have closed streamlines does not necessarily mean you will have fluid blobs rotating about their centers. Odd though it may seem, one can have both rotational and irrotational vortices. An example of a rotational vortex is solid body rotation where we have Ωo ≡ angular velocity uθ = Ωo r, (11.30) An example of an irrotational vortex is the point, or line, vortex, where we have uθ = Γ , 2πr Γ ≡ circulation What is this thing called circulation? Mathematically, it is defined as � Γ = u · ds (11.31) (11.32) C Γ is the sum of the component of velocity tangent to C summing all around C (see figure 11.1). So, if steady, uniform flow, Γ = 0 because all u · ds will cancel one another out (see figure 11.2). Stokes, who was so helpful sorting out our 4th order tensor problems, has a theorem that states: � � u · ds = (� × u) · dA (11.33) C A This related the line integral (about a closed curve) of a vector field to a surface integral of the vector field where the surface is a “capping” surface of the closed curve (see figure 11.3). Stokes theorem is a lot like the divergence theorem (they are linked by exterior calculus), and is sufficiently neat to warrant some time in class. Just like the divergence theorem, we’ll take our capping surface and closed surface, and break each into polygon chunks (see figure 11.4). Assert that circulation around a closed loop is same as the sum of circulations around the polygons. � ⇒ u · ds = ˜ C N � � i=1 4 Ci u · ds (11.34) Figure 11.3: (fig:Lec11Circulation3) A closed surface defined by C, and its “capping” surface, A. Figure 11.4: (fig:Lec11Circulation4) The closed surface defined by C, and its capping surface, A, broken into polygon chunks. I’ll try and convince you of this in the same way as I did for the divergence theorem. See figure 11.5. Consiter the common branch between C1 and C2 . When calculating the component of the circulation in C1 that corresponds to the branch between A and B you have a term like � B u · ds (11.35) A When calculating the component of the circulation in C2 that corresponds to the same branch you get � A � B � A u · ds, since u · ds = − u · ds (11.36) B A B Which means that the contribution to the circulation from the common sides of neighboring poly­ gons will be 0. The fact that there will be canceling contributions on all sides except the sides common with the closed loop confirms that N � � � u · ds = ˜ C i=1 u · ds (11.37) Ci We now apply a trick similar to the one we use for the divergence theorem and multiply and divide the RHS by ΔAi , the area of the ith face. � u · ds C̃ � � N � � 1 = u · ds ΔAi ΔAi Ci i=1 Figure 11.5: (fig:Lec11Circulation5) Note Γ1 = 5 � u · ds1 and Γ2 = (11.38) � u · ds2 . Figure 11.6: (fig:Lec11SolidBodyRotation) Schematic of solid body rotation. Note that uθ = Ωo r and ur = 0. The quantity in the brackets (for ΔAi � 0) is, by definition, the curl of u. The curl is the circulation per area as area goes to zero, and has the direction of normal to that area. � u · ds = lim N →∞,ΔAi →0 ˜ C � u · ds = ˜ C lim N →∞,ΔAi →0 � N � ni · (� × u)Ai (11.39) i=1 (� × u) · dA (11.40) A (Jim – I’m not quite sure how to stack the limits so they do not string out like they do now). Stokes theorem tells us that the line integral of the tangental part of a vector function around a closed path is equal to the surface integral of the normal component of the curl of the vector function over any capping surface of the closed path. Neat! Divergence is the flux per volume while curls is the circulation per area. 11.7 Solid body rotation Where are we now? Solid body rotation is where the velocity is proportional to the radius of the streamlines (see figure 11.6). The vorticity in polar coordinates is given by ωz = ωz = 1 ∂ 1 ∂ur (ruθ ) − r ∂r r ∂θ 1 ∂ 2 1 (r Ωo ) = (2rΩo ) = 2Ωo r ∂r r (11.41) (11.42) This implies that the time it takes a particle to rotate about its own center is the same as it takes the particle to rotate about the center of the SBR. And how does this compare with the circulation? � Γ = u · ds (11.43) C Consider a radius, r �2π Γ= uθ r dθ = uθ r2π = (2Ωo )πr2 = ωz πr2 (11.44) 0 This is just the vorticity times the area, as it should be. Check this with Stokes. � � Γ= (�θ,r × u) · dA = ωz dA = ωz πr2 A (11.45) A As it should be. Note from the picture of SBR (figure 11.6) that fluid elements do not deform. ABCD is not deformed relative to A� B � C � D� . We know that it is the deformation rate of fluid 6 Figure 11.7: (fig:Lec11PotentialVortex1) Schematic of an irrotational (potential) vortex. Here uθ = rc and ur = 0. blobs that leads to viscosity, so since we have no deformation we can conclude that SPR is an inviscid phenomenon – Although viscosity will be important during spin up and at the edge of the SBR. In SBR, the uθ ’s at large r are much greater than the uθ ’s at small r. At first glance one might think we should then have low pressure at large r and high pressure at small r. From Bernoulli: 1 ρ u2θfast + p1 2 = 1 ρ u2θslow + p2 2 (11.46) But remember, ρ 12 u2θ + p + gz is only constant everywhere for inviscid, irrotational flow. SBR is not irrotational, so 1 ρ u2θ + p 2 = constant on a streamline ⇒ constant = f (r) 11.8 (11.47) (11.48) Irrotational (potential) vortex An irrotational (potential) vortex is where the velocity falls off as here is ωz = = = 1 r 1 ∂ 1 ∂ur (ruθ ) − r ∂r r ∂θ 1 ∂ c r ∂r 0 ! r (see figure 11.7). Vorticity (11.49) We have closed, circular streamlines and no vorticity!?! Actually we have zero vorticity everywhere except at the origin. There ωz = 00 , so we’re not really sure what is going on. We can calculate the circulation of a loop enclosing the center of the vortex at some radius r: � Γ = u · ds (11.50) c 2π � Γ = 0 c uθ r dθ = uθ r2π = r2π = 2πc r (11.51) This tells us that Γ is constant – not a function of r as in the case of SBR. It also says that Γ c = 2Γπ ⇒ uθ = 2πr . Consider Stokes’ theorem for a closed curve that includes the origin: � � Γ = u · ds = ω · dA = 2πc (11.52) c A For the area integral to be nonzero, ωz must be nonzero somewhere within the closed curve. Since we know ωz = 0 everywhere except the origin, ωz must be nonzero only there. Then what should 7 Figure 11.8: (fig:Lec11PotentialVortex2) The circulation of any closed curve that does not contain the origin is zero. This is a schematic of such a closed curve. Figure 11.9: (fig:Lec11PotentialVortex3) Deformation of a fluid blob in a potential vortex. the magnitude be? At the origin, dA is infinitely small, so ωz must be infinitely large. A potential vortex is irrotational everywhere except as the origin, where the vorticity is infinite. A delta function of vorticity. The circulation of any closed curve that contains the vortex is the same (Γ = 2πc), but the circulation of and closed curve that does not contain the origin is zero (see figure 11.8 for diagram): � C � D � B � A Γ = u · ds + u · ds + u · ds + u · ds (11.53) A D C A � u · ds + = C D B � u · ds (11.54) B c c r2 Δθ − r1 Δθ r1 r2 = cΔθ − cΔθ = 0 = (11.55) (11.56) (11.57) Note that in the first step, the two integral terms drop out because ur = 0. Consider next the deformation of a fluid blob in a potential vortex as is seen in figure 11.8. There is a deformation of the blob, implying that shear stresses exist. A shear stress implies viscous stresses and viscous stresses imply vorticity. Yet we have no vorticity here. What gives? The rotation of the blob about its own center is exactly balanced by the rotation of the blob about the center of the vortex. Stated in a different manner, the rotation of one axis is exactly balanced by the rotation of the other axis. This, combined with the results from solid body rotation, lead to two important points. • Irrotational flow need not mean there are no viscous stresses (potential vortex). • Absence of viscous stresses need not mean absence of vorticity (solid body rotation). Show film. 11.9 Real vortices I want to briefly say something about combining SBR vortices and potential vortices. “Real vortices” (tornadoes, hurricanes, etc.) behave like SBR near the core, and like a potential vortex 8 Figure 11.10: (fig:Lec11RealVortex) ωz and uz profiles for “real vortices”. Figure 11.11: (fig:Lec11RankineVortex) ωz and uz profiles for a Rankine Vortex. far from the core (see figure 11.10). We can approximate this behavior by defining a vortex with SBR out to some distance R, and potential flow beyond that (see figure 11.11). This is called a Rankine vortex. In read vortices, what sets R? 11.10 Reading for class 12 KC01: 5.4 - 5.5, meaning of (ω · �)u section in 5.6 (Jim - I couldn’t tell if this last bit was from KC or elsewhere). 9